Statistics for Decision Making Basic Inference QM Fall 2003 Instructor: John Seydel, Ph.D.

Student Objectives Use sample data to generate and interpret interval estimates of population parameters Apply the margin of error concept in determining the quality of parameter estimates Work with the Student’s t distribution when developing inferences for quantitative data Determine the required number of sample observations for achieving a desired precision in estimating population/process parameters Compare and contrast the two types of statistical inference Define hypothesis testing and summarize the basic process Discuss errors that can be made in statistical inferences Use sample data to test claims about population parameters

Sampling Distributions (Review) Data TypeParameterEstimatorStdError Quantitative  Qualitative  Note: these estimators are approximately normally distributed; i.e., their sampling distributions are approximately normal

Review of Simple Inference: Estimation Recall: there are only 2 types of inference Estimation (confidence intervals) Hypothesis testing Confidence intervals Parameter ≈ Point Estimate ± Margin of Error Margin of error is based upon confidence level  Margin of error = z-score ∙ standard error  Example, for confidence of 95%: 2 ∙ (s/√n) 2 ∙ (√[(p)(1 - p)/n]) Example (Exercise 7-4)

Estimation, the Procedure Determine parameter needing to be estimated Gather data Calculate appropriate sample statistics Quantitative data: x-bar & s Qualitatative data: p Determine the margin of error Appropriate z-scorez-score Calculate the standard error Calculate: Z ∙ StdErr Put it all together: Parameter = Estimator ± Margin of Error Estimate: (Parameter – Margin) to (Parameter + Margin) Interpret/apply results If appropriate, gather additional data * Note: no sketch! * More on this phases a little later

Interpreting Interval Estimates Strictly: Of all the samples that could be taken from this population, __% of them will result in _____s that are within _____ of the overall population _____. Practically: We are __% confident that the overall population _____ is equal to _____, give or take _____. We can be __% confident that the overall _____ is between _____ and _____. How we typically express findings in the popular press: The survey indicates the the overall _____ of the population is _____. (Margin of error on these findings is _____.) Here’s a good way to look at interval estimates: The margin of error provides an indication of how well the sample statistic estimates the population/process parameter of interest Now, apply to previous examples

Addressing the Quality of the Estimators Again, note that The margin of error provides an indication of how well the sample statistic estimates the population/process parameter of interest Suppose the margin of error is too large; now what? Forget the whole thing? Make up what you want?... ? Of course, not! Go out and get more data How much is enough? Do we just do this over and over again until our precision is sufficient (i.e., margin of error is small enough)? Actually, there’s a way to deal with this...

Determining Necessary Sample Sizes Deriving the needed equations Write a formula for the margin of error Plug in known, required, or estimated values Solve for n Generally requires some sort of pilot sample Demonstration Quantitative data... (Equation 7-8) Qualitative data... (Equation 7-13) The resulting formulae are simpleformulae Applications: Exercises 7-21 and 7-36 Rules of thumb for minimum sample sizes (if normal distribution is to be applied) Quantitative data: n>30 Qualitative data: n  ≥ 5 and n(1-  ) ≥ 5

The Student’s t Distribution Not just a sample size issue Used Always with quantitative data whenever standard deviation is unknown Never, ever with qualitative data! However, it’s so close to the normal once sample sizes get sufficiently large Use special tables Work opposite of normal tables (probability on inside) Involve a third parameter: degrees of freedom (i.e., adjusted sample size) We now call that standard score value t instead of z, but it refers to the same thing Examples (Exercise 7-3)

Intermission: Some Excel Stuff Chart formatting Main title: 12 point Axis titles: 10 point Axis labels: 8 point Printing Use preview Work with setup options  Portrait/landscape  Fit to page  Gridlines, row/column labels Set print area if needed

The Other Kind of Inference: Hypothesis Testing Recall that there are only 2 types of inference Estimation (confidence intervals) Hypothesis testing Starts with a hypothesis (i.e., claim, assumption, standard, etc.) about a population parameter ( , , ,  , distribution,... ) Sample results are compared with the hypothesis Based upon how likely the observed results are, given the hypothesis, a conclusion is made

Hypothesis Testing Start by defining hypotheses Null (H 0 ):  What we’ll believe until proven otherwise  We state this first if we’re seeing if something’s changed Alternate (H A ):  Opposite of H 0  If we’re trying to prove something, we state it as H A and start with this, not the null Then state willingness to make wrong conclusion (  ) Draw a sketch of the sampling distribution Determine the decision rule (DR) Gather data and compare results to DR

Errors in Hypothesis Testing Type I: rejecting a true H 0 Type II: accepting a false H 0 Probabilities  = P(Type I)  = P(Type II) Power = P(Rejecting false H 0 ) = P(No error) Controlling risks Decision rule controls  Sample size controls  Worst error: Type III (solving the wrong problem)! Hence, be sure H 0 and H A are correct

Hypothesis Testing Examples Quantitative data (from text): 3, 4, 5 Qualitative data We haven’t discussed this, but it works the same! Text: 28, 29, 30 Now, about p-values Just another way to express the DR Note: three types of DRDR

Summary of Objectives Use sample data to generate and interpret interval estimates of population parameters Apply the margin of error concept in determining the quality of parameter estimates Work with the Student’s t distribution when developing inferences for quantitative data Determine the required number of sample observations for achieving a desired precision in estimating population/process parameters Compare and contrast the two types of statistical inference Define hypothesis testing and summarize the basic process Discuss errors that can be made in statistical inferences Use sample data to test claims about population parameters

Appendix

Sampling Population Sample Parameter Statistic

Sample Size Formulae Quantitative data (inferences concerning the average): Qualitative data (inferences concerning a proportion) Based on prior estimates: Worst-case scenario:

Inferences: Using the Normal Table in Reverse For inference, we usually start with a probability (i.e., a confidence level or error probability) Then we need to determine the z-score (sometimes called a t-score) associated with that probability Finally, we determined the average or proportion that is z (or t) standard errors away from the base average

Stating the Decision Rule First, note that no analysis should take place before DR is in place! Can state any of three ways Critical value of observed statistic (x-bar or p-hat) Critical value of test statistic (z) Critical value of likelihood of observed result (p-value) Generally, test statistics are used when results are generated manually and p-values are used when results are determined via computer Always indicate on sketch of distribution